Appendices 10.A & 10.B: An Educational Presentation. Presented By: Joseph Ash Jordan Baldwin Justin Hirt Andrea Lance. History of Heat Conduction. Jean Baptiste Biot (1774-1862) French Physicist Worked on analysis of heat conduction
Jean Baptiste Biot
Jean Baptiste Joseph Fourier
transfer that causes a dimensionless
A first approximation of the equations that govern the conduction of heat in a solid rod.
is a proportionality factor called the thermal conductivity and is determined by material properties
where s = specific heat of the material
ρ = density
This is the heat absorption equation.
and has the dimensions of length^2/time and called the thermal diffusivity
where h1 is a proportionality constant
if h1=0, then it corresponds to an insulated end
if h1 goes to infinity, then the end is held at 0 temp.
where, again, h2 is a nonzero proportionality factor
occurring throughout the bar
which is commonly called the generalized heat conduction equation
Let an aluminum rod of length 20 cm be initially at the uniform temperature 25C. Suppose that at time t=0, the end x=0 is cooled to 0C while the end x=20 is heated to 60C, and both are thereafter maintained at those temperatures.
Find the temperature distribution in the rod at any time t
Find the temperature distribution, u(x,t)
2uxx=ut, 0<x<20, t<0
u(0,t)=0 u(20,t)=60, t<0
From the initial equation we find that:
L=20, T1=0, T2=60, f(x)=25
We look up the Thermal Diffusivity of aluminum→2=0.86
Using Equations 16 and 17 found on page 614, we find that
Evaluating cn, we find that
Now we can solve for u(x,t)
Since there is no acceleration in the horizontal direction
However the vertical components must satisfy
where is the coordinate to the center of mass and the weight is neglected
Replacing T with V the and rearranging the equation becomes
Letting , the equation becomes
To express this in terms of only terms of u we note that
The resulting equation in terms of u is
and since H(t) is not dependant on x the resulting equation is
For small motions of the string, it is approximated that
using the substitution that
the wave equation takes its customary form of
The telegraph equation
where c and k are nonnegative constants
cut arises from a viscous damping force
ku arises from an elastic restoring force
F(x,t) arises from an external force
The differences between this telegraph equation and the customary
wave equation are due to the consideration of internal elastic
forces. This equation also governs flow of voltage or current in a
transmission line, where the coefficients are related to the electrical
parameters in the line.
For a vibrating system with more than on significant space coordinate it may be necessary to consider the wave equation in more than one dimension.
For two dimensions the wave equation becomes
For three dimensions the wave equation becomes
Consider an elastic string of length L whose ends are held fixed. The string is set in motion from its equilibrium position with an initial velocity g(x). Let L=10 and a=1. Find the string displacement for any time t.
From equations 35 and 36 on page 631, we find that
Solving for kn, we find:
Now we can solve for u(x,t)